Using Multiple Outcomes to Adjust Standard Errors for Spatial Correlation

Abstract

Empirical research in economics often examines the behavior of agents located in a geographic space. In such cases, statistical inference is complicated by the interdependence of economic outcomes across locations. A common approach to account for this dependence is to cluster standard errors based on a predefined geographic partition. A second strategy is to model dependence in terms of the distance between units. Dependence, however, does not necessarily stop at borders and is typically not determined by distance alone. This paper introduces a method that leverages observations of multiple outcomes to adjust standard errors for cross-sectional dependence. Specifically, a researcher, while interested in a particular outcome variable, often observes dozens of other variables for the same units. We show that these outcomes can be used to estimate dependence under the assumption that the cross-sectional correlation structure is shared across outcomes. We develop a procedure, which we call Thresholding Multiple Outcomes (TMO), that uses this estimate to adjust standard errors in a given regression setting. We show that adjustments of this form can lead to sizable reductions in the bias of standard errors in calibrated U.S. county-level regressions. Re-analyzing nine recent papers, we find that the proposed correction can make a substantial difference in practice.